Finite Group Databasebeta

← #6508773

Subgroups — #6508773

Order 512.

1922 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic
4#34245C211yesmaximal, characteristic
5#28124C212yesmaximal, complemented
6#28121C212yesmaximal, complemented
7#27653C212yesmaximal, complemented
8#27648C212yesmaximal, complemented
9#25215C214yesmaximal, complemented
10#25201C214yesmaximal, complemented
11#24588C214yesmaximal, complemented
12#24540C214yesmaximal, complemented
13#21932C211yesmaximal, characteristic
14#21768C211yesmaximal, characteristic
15#20699C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3189—28no—
20#3095—28no—
21C2×#484C2212yes—
22C2×#484C2212yes—
23C2×#484C2212yes—
24C2×#485C2214yes—
25C2×#483C2212yes—
26C2×#467C2214yes—
27C2×#451C2212yes—
28C2×#451C2212yes—
29C2×#428C2212yes—
30C2×#419C2212yes—
31C2×#407C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33#2499—21no—
34#2499—21no—
35#2499—21no—
36#2499—21no—
37#2499C2214yes—
38#2499C2214yes—
39C2×#397—22no—
40#2368—22no—
41#2362C2212yes—
42#2362C2212yes—
43#2344C2212yes—
44#2342—22no—
45Q8×C22⋊C4C2212yes—
46#2332—21no—
47#2332—21no—
48#2313—21no—
49C4×C22⋊Q8C2212yes—
50C4×C22⋊Q8C2212yes—
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#387—22no—
54C2×#384C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2280C2211yescharacteristic
57#2274—21no—
58C2×C4×C22⋊C4C2211yescharacteristic
59#2085C2214yescomplemented
60#2083C2214yescomplemented
61#2074C2214yescomplemented
62#2072C2214yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2056C2214yescomplemented
66#2056C2214yescomplemented
67#1949C2214yescomplemented
68#1947C2214yescomplemented
69#1928C2214yescomplemented
70#1927C2214yescomplemented
71#1894C2214yes—
72#1878C2214yescomplemented
73#1878C2214yescomplemented
74#1873C2212yes—
75#1873C2212yes—
76#1851C2214yes—
77#1850C2214yes—
78#1825C2214yes—
79#1824C2212yes—
80#1824C2212yes—
81#1823C2212yes—
82#1805C2214yescomplemented
83#1803C2214yescomplemented
84#1797C2212yes—
85#1797C2212yes—
86#1787C2212yes—
87#1735C2212yes—
88#1735C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×Q16C2314yes—
93C22×M16C2312yes—
94C23×C8C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8D418yes—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×#117—21no—
110C2×#117—21no—
111C2×#117—22no—
112C2×#117—24no—
113C2×#117—24no—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4D412yes—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#485—216no—
133#484—28no—
134#484—28no—
135#484—28no—
136#483—28no—
137#467—216no—
138#451—28no—
139#451—28no—
140#428—28no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2314yes—
146#419—28no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153#407—24no—
154C2×C22⋊C8—24no—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159#397—24no—
160#397—24no—
161#397—24no—
162#397—24no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391C2312yes—
170#391C2312yes—
171#391C2314yes—
172#391C2314yes—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—24no—
186#387—24no—
187#387—24no—
188#387—24no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384C2314yes—
194#382—21no—
195#382—21no—
196#382C2314yes—
197#381—21no—
198#381—22no—
199#380—22no—
200#380C2312yes—
201#380C2312yes—
202#379—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D411yescharacteristic
215C4×C22⋊C4D411yescharacteristic
216C4×C22⋊C4D411yescharacteristic
217C4×C22⋊C4D411yescharacteristic
218C4×C22⋊C4C2312yes—
219C4×C22⋊C4C2312yes—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C25C2×D411yescharacteristic
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8C2×D414yes—
243C22×Q8C2412yes—
244C22×Q8C2412yes—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—42no—
252C23×C4—42no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D414yes—
259C23×C4#4114yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×Q16—28no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×M16—24no—
267C2×M16—42no—
268C2×M16—42no—
269C22×C8—24no—
270C22×C8—42no—
271C22×C8—42no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8C2×D418yes—
285C22⋊Q8C2×D418yes—
286C4×Q8—44no—
287C4×Q8—44no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—48no—
293#117—44no—
294#117—44no—
295#117—42no—
296#117—42no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2412yes—
363#107—28no—
364#107—28no—
365#106—28no—
366C4⋊C8—44no—
367C4⋊C8—44no—
368#103—28no—
369#103—28no—
370#103—28no—
371#103—28no—
372#103—44no—
373#103—44no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377C22⋊C8—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—24no—
399C24—42no—
400C24—41no—
401C24—41no—
402C24#12114yes—
403C24C22×D411yescharacteristic
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—48no—
417C2×Q8—42no—
418C2×Q8—81no—
419C2×Q8—81no—
420C2×Q8—84no—
421C2×Q8—84no—
422C2×Q8—84no—
423C2×Q8C22×D414yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4#12114yes—
467C22×C4#12114yes—
468C22×C4#12114yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2511yescharacteristic
483Q16—88no—
484M16—48no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—84no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4C22×D414yes—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—81no—
529C22⋊C4—81no—
530C42—24no—
531C42—24no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—44no—
537C42—44no—
538C42—44no—
539C42—44no—
540C42—81no—
541C42—81no—
542C42—82no—
543C42—82no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—28no—
556C23—44no—
557C23—44no—
558C23—84no—
559C23—84no—
560C23C2×#12112yes—
561C23C2×#12112yes—
562C23C2×#12112yes—
563C23#57814yes—
564C23C23×D411yescharacteristic
565Q8—28no—
566Q8—44no—
567Q8—44no—
568Q8—44no—
569Q8—84no—
570Q8—84no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4C2×#12112yes—
627C2×C4C2×#12112yes—
628C2×C4C2×#12112yes—
629C2×C4#57514yes—
630C2×C4C23×D411yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4#58512yes—
633C2×C4#58512yes—
634C8—44no—
635C8—44no—
636C22—24no—
637C22—24no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—88no—
651C22#304414yes—
652C22C22×#12111yescharacteristic
653C22#348511yescharacteristic
654C22#348511yescharacteristic
655C22C2×#57512yes—
656C22C2×#57812yes—
657C4—24no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—44no—
665C4—44no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4#348512yes—
673C4#349112yes—
674C2—24no—
675C2—42no—
676C2—42no—
677C2#3544711yescharacteristic
678C2#6041511yescharacteristic
679C2C2×#304411yescharacteristic
680C2#6173012yes—
681C2#6173812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #28124C2semidirect
6. #28121C2semidirect
7. #27653C2semidirect
8. #27648C2semidirect
9. #25215C2semidirect
10. #25201C2semidirect
11. #24588C2semidirect
12. #24540C2semidirect
59. #2085C22semidirect
60. #2083C22semidirect
61. #2074C22semidirect
62. #2072C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
65. #2056C22semidirect
66. #2056C22semidirect
67. #1949C22semidirect
68. #1947C22semidirect
69. #1928C22semidirect
70. #1927C22semidirect
72. #1878C22semidirect
73. #1878C22semidirect
82. #1805C22semidirect
83. #1803C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#31892
C2×#30952
C2×#24992
#342452
#281242
#281212
#276532
#276482
#252152
#252012
#245882
#245402
#219322
#217682
#206992